To prove our main result Theorem REF , we shall follow the framework of Hochman and Shmerkin [1]} to build up a relevant CP-chain and estimate its local entropy averages, then use Marstrand's projection theorem to find the right dimension. The main difficulty of extending their methods to our setting comes from the possible overlaps in the IFSs and the random extension due to the Mandelbrot cascade action, which we shall explain here.
